[rohanarun]

Conditional κ = 0.001734846264120199151475 − 10⁻⁴⁶ − 10⁻⁷⁰, +0.00176% over #3251 (0.001734815770385098563484). This takes the whole remaining window to the R438 bit endpoint. Draft: full verification is running.

	complex root lower	bit	binds	κ
#3251	0.0017348157703850985634843	R438 0.001734846264120199151475	complex	0.001734815770385098563484 − …
this	0.0017348651784727109982331	R438 0.001734846264120199151475	bit	0.001734846264120199151475 − …

Change

After #3251, 149,514 rank-1 idle remainders remain:

• 107,356 fail Idle pair exchanges improve the #1094 κ candidate #1252's translation rule, so no receiver can take them.
• 39,558 conflict with every unhosted child of their group, and the group's other children are all hosted.
• 2,600 conflict with every unhosted child.

Relaying through an unhosted child was tested and found nothing legal.

New in this PR is a fifth phase: an idle-remainder axis may move into the remainder of another partially hosted bin in the same (replica, port). The axis must commute with that bin's full recipe frame (hosted prefix plus remainder), with all stage X/Y translations and the terminal Y translation, and with both terminal translations. Receivers are capped at 51, and no child both donates and receives.

This adds 2,374 moves, 437,978 in total, mostly merging pairs of rank-1 remainders into rank-2 children. Rank mass is unchanged at 65088·740.

Verification

research/rank7-idle-hosted-receivers-r438/verify.py is #3251's independent verifier, extended for hosted receivers. It re-derives:

• every move's legality, including orthogonality to the receiver's hosted prefix;
• the profile and rank mass;
• the exact complex root bracket;
• the R438 endpoint;
• κ = min(floor₂₄(complex), bit) − …, with the bit side binding.

Nine mutation controls are rejected, including a prefix-conflicting hosted receiver. scripts/research_package.py passes locally under Python 3.12.

Scope

This adds a new hypothesis: hosted remainders may receive axes, extending #3190's hosted-donor hypothesis. No hosted-controller argument is given.

Not rerun yet: the native admission, the overlay-aware controller audit, and the outer release.

The result is conditional on R438 (#1043 is still open) and on the full stack #3251 → #3200 → #3198 → #3190 → #3184 → #3148 → #3133.

Credits: Gonzalo-Mier (#1252), Guéorgui Maissouradze, Jeremy Diviney (#3148), dcharb78 (#3133), Marius-Paul Dumitrean (#2979), Ali Romman (#1043), Sagar Thakrar (#2423), DreamingOfClouds, eumemic, and OpenCode (Space bunny free) for ideas in the stacked predecessors. Prepared by Rohan Arun with Anthropic Claude assistance.

🤖 Generated with Claude Code

https://claude.ai/code/session_011S35E2acyQDaVRjPRw4TDy

[ChaloupeMusic]

FINDING / no new κ. This chain's complex root comes from #3934 and #3955, which stack on #3251, retracted on 2026-10-18 after merged #3296/#3311. #3934's MOVES contain all 435,604 #3251 moves, and #3934 and #3955 both contain #3296's port-0 witness. With #3187/#3296's exact operator replay, each of #3934's 2,374 hosted-receiver moves changes the operator at its own owner and port (2,274 configurations, 140 owners, none preserved). At owner 0, all 55 identity-selection #3955 ports change it; the other 57 need a cell-to-axis map that #3955 does not give. Without these moves the complex side falls back to #3148 (0.0017298), which binds. Package and verifier: #4046.

[rohanarun]

Retracted, closing. This draft belongs to the #3934 → #3959 chain, which is built on #3251's idle-axis moves. Those were retracted on 2026-10-18 because #3190's donations change the exact eight-stream operator (see #3296/#3311/#3187). Native admission, the controller audit and the outer release never ran. #3995 and #4046 show the pooled move is rejected. The κ claimed here is not valid. Prepared by Rohan Arun with Anthropic Claude assistance.